Brier, who shot 3-under-par 68 in the closing 18 holes, finished 18-under 266, three strokes better than Denmark's Soren Hansen (65).
"To make history as the first Austrian to win on The European Tour is fantastic," said Brier, who had a hole-in-one and an eagle during the tournament. "I am proud to have done it. To do it here around my home track is just unbelievable."
Dyson was 14-under 270, Green finished 13-under 271, and Murphy was 10-under 274.
Scotland's Colin Montgomerie (279) fell out of contention when he dropped five shots in four holes.
Dyson was tied with Brier after two rounds.